504 THE THEORY OF SCREWS. 



compendious law should be discovered, which connected the impulsive screw 

 with the instantaneous screw, their experiments would indeed be endless. Was it 

 likely that such a law could be found was it even likely that such a law existed ? 

 Mr Querulous decidedly thought not. He pointed out how the body was 

 of the most hopelessly irregular shape and mass, and how the constraints were 

 notoriously of the most embarrassing description. It was, therefore, he thought, 

 idle to search for any geometrical law connecting the impulsive screw and the 

 instantaneous screw. He moved that the whole inquiry be abandoned. These 

 sentiments seemed to be shared by other members of the committee. Even the 

 resolution of the chairman began to quail before a task of infinite magnitude. A 

 crisis was imminent when Mr Anharmonic rose. 



Mr Chairman, he said, Geometry is ever ready to help even the most 

 humble inquirer into the laws of Nature, but Geometry reserves her most gracious 

 gifts for those who interrogate Nature in the noblest and most comprehensive spirit. 

 That spirit has been ours during this research, and accordingly Geometry in this 

 our emergency places her choicest treasures at our disposal. Foremost among these 

 is the powerful theory of homographic systems. By a few bold extensions we 

 create a comprehensive theory of homographic screws. All the impulsive screws 

 form one system, and all the instantaneous screws form another system, and 

 these two systems are homographic. Once you have realised this, you will find 

 your present difficulty cleared away. You will only have to determine a few pairs 

 of impulsive and instantaneous screws by experiment. The number of such pairs 

 need never be more than seven. When these have been found, the homography is 

 completely known. The instantaneous screw corresponding to every impulsive 

 screw will then be completely determined by geometry both pure and beautiful. 

 To the delight and amazement of the committee, Mr Anharmonic demonstrated 

 the truth of his theory by the supreme test of fulfilled prediction. When the 

 observations had provided him with a number of pairs of screws, one more than 

 the number of degrees of freedom of the body, he was able to predict with in 

 fallible accuracy the instantaneous screw corresponding to any impulsive screw. 

 Chaos had gone. Sweet order had come. 



A few days later the chairman summoned a special meeting in order to hear 

 from Mr Anharmonic an account of a discovery he had just made, which he 

 believed to be of signal importance, and which he was anxious to demonstrate by 

 actual experiment. Accordingly the committee assembled, and the geometer pro 

 ceeded as follows : 



You are aware that two homographic ranges on the same ray possess two 

 double points, whereof each coincides with its correspondent ; more generally when 

 each point in space, regarded as belonging to one homographic system, has its 

 correspondent belonging to another system, then there are four cases in which a 

 point coincides with its correspondent. These are known as the four double points, 

 and they possess much geometrical interest. Let us now create conceptions of an 

 analogous character suitably enlarged for our present purpose. We have dis 

 covered that the impulsive screws and the corresponding instantaneous screws form 

 two homographic systems. There will be a certain limited number (never more 



